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P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

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Abstract

In this paper we use P 1-nonconforming quadrilateral finite volume methods with interpolated coefficients to solve the semilinear elliptic problems. Two types of control volumes are applied. Optimal error estimates in H 1-norm on the quadrilateral mesh and superconvergence of derivative on the rectangular mesh are derived by using the continuity argument, respectively. In addition, numerical experiments are presented adequately to confirm the theoretical analysis and optimal error estimates in L 2-norm is also observed obviously. Compared with the standard Q 1-conforming quadrilateral element, numerical results of the proposed finite volume methods show its better performance than others.

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References

  1. Acosta, G., Duran, R.: Error estimates for Q 1 isoparametric elements satisfying a weak angle condition. SIAM J. Numer. Anal. 38(4), 1073–1088 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bank, R.E., Rose, D.J.: Some error estimates for the box methods. SIAM J. Numer. Anal. 24(4), 777–787 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bramble, J.H.: A second-order finite difference analog of the first biharmonia boundary value problem. Numer. Math. 9(4), 236–249 (1996)

    MathSciNet  Google Scholar 

  4. Cai, Z.Q.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chatzipantelidis, P.: Finite volume methods for elliptic PDE’s: a new approach. Math. Model. Numer. Anal. 3, 307–324 (2002)

    Article  MathSciNet  Google Scholar 

  6. Chatzipantelidis, P., Ginting, V., Lazarov, R.D.: A finite volume element method for a nonlinear elliptic problem. Numer. Linear Algebra Appl. 12(5), 515–545 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, C.M., Huang, Y.Q.: High Accuracy Theory of Finite Element Methods. Hunan Science, Changsha (1995). (in Chinese)

    Google Scholar 

  8. Chen, C.M., Larson, S., Zhang, N.Y.: Error estimates of optimal order for finite element methods interpolated coefficients for the nonlinear heat equation. IMA J. Numer. Anal. 9, 507–524 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, L.: A new class of high order finite volume methods for second order elliptic equations. SIAM J. Numer. Anal. 47(6), 4021–4043 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chou, S.H.: Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comput. 66, 85–104 (1997)

    Article  MATH  Google Scholar 

  11. Chou, S.H., He, S.N.: On the regularity and uniformness conditions on quadrilateral grids. Comput. Methods Appl. Mech. Eng. 191, 5149–5158 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chou, S.H., Kwak, D.Y., Kim, K.Y.: A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: The overlapping covolume case. SIAM J. Numer. Anal. 39(4), 1170–1196 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duran, R., Lombardi, A.: Error estimates on anisotropic Q 1 elements for functions in weighted Sobolev spaces. Math. Comput. 74(252), 1679–1706 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. El Alaoui, L.: An adaptive finite volume box scheme for solving a class of nonlinear parabolic equations. Appl. Math. Lett. 22, 291–296 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eymard, R., Gallouet, T., Herbin, R.: Finite Volume Methods. In: Handbook of Numerical Analysis VII, pp. 713–1020. North-Holland, Amsterdam (2000)

    Google Scholar 

  16. Feng, X.L., Kim, I., Nam, H., Sheen, D.: Locally stabilized P 1-nonconforming quadrilateral and hexahedral finite element method for the Stokes equations. J. Comput. Appl. Math. 236(5), 714–727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frehse, J., Rannacher, R.: Asymptotic L ∞-error estimate for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal. 15, 418–431 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grajewski, M., Hron, J., Turek, S.: Numerical analysis for a new nonconforming linear finite element on quadrilaterals. J. Comput. Appl. Math. 193, 38–50 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. He, Y.N., Feng, X.L.: H 1-stability and convergence of the FE, FD and FV methods for the Poisson equation. Preprint

  20. Hu, J., Shi, Z.C.: Constrained quadrilateral nonconforming rotated Q 1 element. J. Comput. Math. 23(5), 561–586 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Lamichhane, B.P.: Inf-sup stable finite-element pairs based on dual meshes and bases for nearly incompressible elasticity. IMA J. Numer. Anal. 29, 404–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Larson, S., Thomee, V., Zhang, N.Y.: Interpolation of coefficients and transformation of dependent variable in element methods for the nonlinear heat equation. Math. Methods Appl. Sci. 11, 105–124 (1989)

    Article  MathSciNet  Google Scholar 

  23. Li, R.H., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  24. Li, Y.H., Li, R.H.: Generalized difference methods on arbitrary quadrilateral networks. J. Comput. Math. 17, 653–672 (1999)

    MathSciNet  MATH  Google Scholar 

  25. Liu, H.P., Yan, N.N.: Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations. Adv. Comput. Math. 29, 375–392 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lv, J.L., Li, Y.H.: L 2 error estimate of the finite volume element methods on quadrilateral meshes. Adv. Comput. Math. 33, 129–148 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Man, H.Y., Shi, Z.C.: P 1-nonconforming quadrilateral finite volume element method and its cascadic multigrid algorithm. J. Comput. Math. 24, 59–80 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Mao, S.P., Chen, S.C.: A new nonconforming double set parameter finite element. In: Proceedings of the Seventh China-Japan Seminar on Numerical Mathematics, ZhangjiaJie (2004)

    Google Scholar 

  29. Mao, S.P., Chen, S.C.: A quadrilateral nonconforming finite element for linear elasticity problem. Adv. Comput. Math. 28, 81–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mao, S.P., Chen, S.C.: Convergence and superconvergence of a non-conforming finite element for Stokes problem. J. Numer. Math. 14, 83–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mao, S.P., Chen, S.C., Sun, H.X.: A quadrilateral, anisotropic, superconvergent nonconforming double set parameter element. Appl. Numer. Math. 56, 937–961 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mao, S.P., Nicaise, S., Shi, Z.C.: On the interpolation error estimates for Q 1 quadrilateral finite elements. SIAM J. Numer. Anal. 47(1), 467–486 (2008)

    Article  MathSciNet  Google Scholar 

  33. Park, C.: A study on locking phenomena in finite element methods. Ph.D. thesis, Department of Mathematics, Seoul National University, Seoul, Korea (2002). Available at http://www.nasc.snu.ac.kr/cpark/papers/phdthesis.ps.gz

  34. Park, C., Sheen, D.: P 1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41(2), 624–640 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schmidt Kiel, T.: Box schemes on quadrilateral meshes. Computing 51, 271–292 (1993)

    Article  MathSciNet  Google Scholar 

  36. Wang, T.K.: A mixed finite volume element method based on rectangular mesh for biharmonic equations. J. Comput. Appl. Math. 172, 117–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xiong, Z.G., Chen, C.M.: Superconvergence of triangular quadratic finite element method with interpolated coefficients for nonlinear elliptic problem. Acta Math. Sci. Ser., 26(2), 174–182 (2006). (in Chinese)

    MathSciNet  MATH  Google Scholar 

  38. Xiong, Z.G., Chen, C.M.: Superconvergence of rectangular finite element with interpolated coefficients for semilinear elliptic problem. Appl. Math. Comput. 181, 1577–1584 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xiong, Z.G., Chen, C.M.: Superconvergence of continuous finite elements with interpolated coefficients for initial value problems of nonlinear ordinary differential equation. Numer. Math. J. Chin. Univ. 16(1), 37–44 (2007)

    MathSciNet  Google Scholar 

  40. Xiong, Z.G., Chen, Y.P.: A rectangular finite volume element method for a semilinear elliptic equation. J. Sci. Comput. 36, 177–191 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xiong, Z.G., Chen, Y.P.: Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations. Comput. Methods Appl. Mech. Eng. 196, 3798–3804 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, J.C., Zou, Q.S.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numer. Math. 111(3), 469–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yang, M.: A second-order finite volume element method on quadrilateral meshes for elliptic equations. Modél. Math. Anal. Numér. 40(6), 1053–1067 (2006)

    Article  MATH  Google Scholar 

  44. Zhang, H.Q., Wang, M.: The Mathematical Theory of Finite Elements. Science Press, Beijing (1991). (in Chinese)

    Google Scholar 

  45. Zhu, P., Li, R.H.: Generalized difference methods for second order elliptic partial differential equations (II)-quadrilateral grids. Numer. Math. J. Chin. Univ. 4, 360–375 (1982)

    MATH  Google Scholar 

  46. Zlamal, M.: A finite element solution of the nonlinear heat equation. RAIRO Model. Anal. Numer 14, 203–216 (1980)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P.R. China

    Xinlong Feng, Rongfei Li, Yinnian He & Demin Liu

  2. Faculty of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Yinnian He

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  1. Xinlong Feng
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  2. Rongfei Li
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Correspondence to Yinnian He.

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Feng, X., Li, R., He, Y. et al. P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations. J Sci Comput 52, 519–545 (2012). https://doi.org/10.1007/s10915-011-9557-4

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  • DOI: https://doi.org/10.1007/s10915-011-9557-4

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